A turbidity measuring device according to the four-beam, alternating light principle includes at least two light sources and at least two receivers, wherein four measuring paths are defined between the two measuring sources and the two receivers, via which the light emitted by the light sources reaches the receivers; wherein, on at least two measuring paths, the light reaches the receiver through scattering. In general, the signal Sij (T) of the receiver Rj, which receives light emitted by the light source Li following an interaction with a measured medium, is given by Equation 1.
                                          S            ij                    ⁡                      (            T            )                          =                              I            i                    ·                      c            ij                    ·          T          ·                      ⅇ                          -                                                T                  ·                                      X                    ij                                                  λ                                                                        (        1        )            
In such a case, Ii is the intensity of the emitted light; Cij a is constant, which is dependent on the geometrical boundary conditions of the turbidity measuring device and the scattering properties of the turbidity-causing material, Xij is the measuring path length in the measured medium between the light source Li and the receiver Rj, and λ is a coefficient, which describes the scattering and absorption characteristics of the turbidity-causing material with regard to the radiated light, wherein the turbidity-causing material is present in a concentration T.
In order to eliminate the influence of variable device parameters such as, for example, the intensity of the radiated light I1, I2 and transmission characteristics of windows, the measured variable FAL(T)—defined in Equation 2—is introduced (the acronym FAL comes from Four-beam, Alternating Light), the explicit representation of which is given in Equation 3.
                              FAL          ⁡                      (            T            )                          =                                                            S                11                            ⁡                              (                T                )                                      ·                                          S                22                            ⁡                              (                T                )                                                                                        S                12                            ⁡                              (                T                )                                      ·                                          S                21                            ⁡                              (                T                )                                                                        (        2        )                                          FAL          ⁡                      (            T            )                          =                                                            c                11                            ·                              c                22                                                                    c                12                            ·                              c                21                                              ·                      ⅇ                          T              ⁢                                                          ⁢                                                                    X                    12                                    +                                      X                    21                                    -                                                            X                      11                                        ⁢                                                                                  ⁢                                          X                      22                                                                      λ                                                                        (        3        )            
It should be recognized here, that the measured variable FAL(T) is independent of the radiated intensities, and the dependence of the concentration T on the turbidity-causing material is present only in the exponential function.
If one furthermore assumes a symmetry in the construction of the turbidity measuring device, this thus meaning that c11=c22 and c12=c21, as well as X11=X22=Xdirect and X12=X21=Xindirect, then the measured variable FAL(T) can be represented in the form of Equation 4:
                                          FAL            ⁡                          (              T              )                                =                      c            ·                          ⅇ                                                2                  ·                  T                                ⁢                                                                  ⁢                                                                            X                      indirect                                        -                                          X                      direct                                                        λ                                                                    ,                            (        4        )            wherein c represents the quotient of the coefficients.
In FIG. 4a, an example of a curve of an FAL-signal, represented as a function of the content T of the turbidity-causing material (TCM), is presented as solid line. For high concentrations of turbidity-causing material, the FAL-signal is a good signal to evaluate, and directly enables an association between signal value and content of turbidity-causing material. In the case of low concentrations, below the maximum of the signal of the individual measurement channels Sij, the FAL-signal has, however, weaknesses, which make an exact determining of the concentration of turbidity-causing material difficult, because (as is presented in Equation 5), for low concentrations, the variable FAL(T) converges toward the constant C, so that a dependence on the concentration of turbidity-causing material is practically no longer given.
                                          FAL            ⁡                          (              T              )                                ⁢                      ⟶                          T              ⁢                              <<                                  λ                                      Δ                    ⁢                                                                                  ⁢                    x                                                                                ⁢          c                ·                              (                          1              +                                                2                  ·                  T                                ⁢                                                                  ⁢                                                      Δ                    ⁢                                                                                  ⁢                    X                                    λ                                                      )                    ⟶          c                                    (        5        )            (In such a case, ΔX:=Xdirect−Xindirect)
The independence of the FAL-signal from the measured variable is again made clear in FIG. 4b, where it is logarithmically plotted versus small values for the content of turbidity-causing material.